52 1. Preparatory material

a singular value decomposition with singular values σ2 ≥ · · · ≥ σp ≥ 0 and

singular vectors u2,...,up ∈ u1

⊥,

v2,...,vp ∈ v1

⊥

with the stated properties.

By construction we see that σ2,...,σp are less than or equal to σ1. If we

now adjoin σ1,u1,v1 to the other singular values and vectors we obtain the

claim.

Exercise 1.3.16. Show that the singular values σ1(A) ≥ · · · ≥ σp(A) ≥ 0

of a p × n matrix A are unique. If we have σ1(A) · · · σp(A) 0, show

that the singular vectors are unique up to rotation by a complex phase.

By construction (and the above uniqueness claim) we see that σi(UAV )

= σi(A) whenever A is a p × n matrix, U is a unitary p × p matrix, and V is

a unitary n × n matrix. Thus the singular spectrum of a matrix is invariant

under left and right unitary transformations.

Exercise 1.3.17. If A is a p × n complex matrix for some 1 ≤ p ≤ n, show

that the augmented matrix

˜

A :=

0 A

A∗

0

is a p+n×p+n Hermitian matrix whose eigenvalues consist of ±σ1(A),...,

±σp(A), together with n − p copies of the eigenvalue zero. (This generalises

Exercise 2.3.17.) What is the relationship between the singular vectors of A

and the eigenvectors of

˜?

A

Exercise 1.3.18. If A is an n×n Hermitian matrix, show that the singular

values σ1(A),...,σn(A) of A are simply the absolute values |λ1(A)|,...,

|λn(A)| of A, arranged in descending order. Show that the same claim

also holds when A is a normal matrix (that is, when A commutes with

its adjoint). What is the relationship between the singular vectors and

eigenvectors of A?

Remark 1.3.10. When A is not normal, the relationship between eigen-

values and singular values is more subtle. We will discuss this point in later

sections.

Exercise 1.3.19. If A is a p × n complex matrix for some 1 ≤ p ≤ n,

show that

AA∗

has eigenvalues

σ1(A)2,...,σp(A)2,

and

A∗A

has eigenval-

ues

σ1(A)2,...,σp(A)2

together with n − p copies of the eigenvalue zero.

Based on this observation, give an alternate proof of the singular value de-

composition theorem using the spectral theorem for (positive semi-definite)

Hermitian matrices.

Exercise 1.3.20. Show that the rank of a p × n matrix is equal to the

number of non-zero singular values.